Dr. Paul Becker
Assistant Professor of Mathematics
Penn State Erie, The Behrend College

Research Interests

Algebraic combinatorics:

Combinatorics is the field of mathematics which studies arrangements of objects into useful patterns.
Algebraic combinatorics applies algebraic techniques to answer fundamental combinatorial questions.
Is a particular complex pattern possible? If the pattern is possible, how can it be constructed?

 Design theory assigns elements from a set to (possibly overlapping) categories.  The elements are called
points or varieties, while the categories are called blocks. A collection of blocks of size k selected from
a set of v points is a block design. It is incomplete if k < v.  

Most interesting designs are subject to regularity conditions. Symmetric designs, for example, have
equal numbers of points and blocks. Balanced block designs require every pair of points to
occur together in exactly λ blocks.

Block designs may be viewed as binary incidence matrices. Each column the matrix represents a point,
while each row represents a block. A nonzero entry in position (i,j) indicates that point i is a member
of block i.  Incidence matrices for symmetric balanced incomplete block designs satisfy:
MM^T = nI + λ J. Specific solutions to this equation include weighing matrices, Hadamard
matrices, perfect ternary arrays, and difference sets. These designs have direct applications in digital
communications, particularly in the construction of error-correcting codes.

Click here to read more about algebraic constructions for block designs.

Representations of finite groups:

A group is a set of elements, G, together with a reasonable binary operation - a method for
combining two elements to produce a third. For example, we say that the set
C₉ = { e, x, x², ... , x⁸}, forms a group under the operation ``multiply, mod 9." This
means that x x³ = x⁴,  while x⁵ x⁴ = x⁹ = x⁰. We generally denote x⁰ by e, and call it the
identity element. The operation of a group must satisfy certain requirements; for
example, the identity must be unique, and every element g must have an inverse such
that g g^-1 = g^-1g = e.  

In principle, every finite group is isomorphic to a group of matrices.  In other words, we
can replace each element of a group by a specific matrix, and replace the group operation by
normal matrix multiplication.  This replacement is called a representation of the group.
In practice, a specific group admits many possible representations.  I recently published a
description of convenient, easily explained representations for semi-direct product groups. 
A copy of that article is available here:  American Math Monthly article, May 2005.  
 

Mathematics education:

I am interested in mathematics education beyond the usual college classroom.  My interests
can be roughly categorized as: secondary mathematics education; teaching with technology;
and collaborative research with undergraduates.

With Paul Olson (Penn State - Erie), I have presented several teacher's workshops on
combinatorics in secondary classrooms. 

I have written collections of computer laboratory exercises for college calculus and linear
algebra courses.  I update these yearly, and use them in a variety of my courses.  I am
also developing a collection of computer laboratories for abstract algebra courses.

I have supervised several mathematical research projects with Penn State undergraduates.
These students have presented their results at regional research conferences.  Three of these
students are co-authors on published and submitted manuscripts.   

Links: